How do I make a free form tail animation?
Parent an UV-sphere to the dog (?...sorry) sized and positioned that way, that
its "north pole" touches the end of the dog and its south pole is just a little 
further away from the dog as the tip of its tail. Or in other words: The diameter
of the sphere is roughly a little longer than the dogs tail.
Parent the sphere to the armature in object mode.
Vertex parent an empty to the "south pole vertex" of the sphere.
Attach an IK-solver to the tail and as target insert the empty.
Adjust the chain length accordingly.
Add a softbody simulation to the sphere.
Make the north pole the goal for pinning the sphere at a stiffniss of "1".
Play with the other parameters of the softbody simulation.
By the way: When jumping around from one frame to a totally different frame
the cache of the softbody simulation can get corrupted and blender not alway notice
that ... which leads to very weird results or none at all.
The easiest way to clear the cache is to set the number of the last stored from
before the number of the first frame temporarily.
Don't forget to set both values to the length of your animation at least.